The purpose of this study is to investigate the reasons and backgrounds of drawing auxiliary lines in the proof of geometry. In most of proofs in geometry, drawing auxiliary lines provide important clues, thus they play a key role in deductive proof. However, many student tend to have difficulties of drawing auxiliary lines because there seems to be no general rule to produce auxiliary lines. To alleviate such difficulties, informal activities need to be encouraged prior to draw auxiliary lines in rigorous deductive proof. Informal activities are considered to be contrasting to deductive proof, but at the same time they are connected to deductive proof because each in formal activity can be mathematically represented. For example, the informal activities such as fliping and superimposing can be mathematically translated into bisecting line and congruence. To elaborate this idea, some examples from the middle school mathematics were chosen to corroborate the relation between informal activities and deductive proof. This attempt could be a stepping stone to the discussion of how to teach auxiliary lines and deductive reasoning.

The purpose of this thesis is, through analysing the characteristics of the definitions in Korean school mathematics textbooks, to explore the levels of them. Definitions use din academic mathematics are rigorous. But they should be transformed into various types, which are presented in school mathematics textbooks, with didactical purposes. In this thesis we investigated such types of transformation. With the result of this investigation we tried to identify the levels of the definitions in school mathematics textbooks. We tried to construct, with consideration about methods of definition, frame for analysing the types of the definitions in school mathematics. Methods of definition are classified as connotative method, denotative method, and synonymous method. Especially we identified that connotative method contains logical definition, genetic definition, relational definition, operational definition, and axiomatic definition. With these analyses we made a frame for investigating the characteristics of the definitions in school mathematics textbooks. With this frame we identified concrete types of transformations of methods of definition. We tried to analyse this result with van Hieles' theory about let·els of geometry learning and the mathematical language levels described by Freudenthal, and identify the levels of definitions in school mathematics. We showed the levels of definitions in the geometry area of the Korean school mathematics.

The purpose of this study is to investigate various perspectives on the approaching to school algebra. School algebra plays an important role in school mathematics, but many students have been in difficulties for the loaming of school algebra. This problem has been continued for the long time. This paper supposed that this problem may be caused by the approaching stage to the school algebra. In order to investigate this problem, we first described the definition of algebra in chapterII. And in chapterIII, we divided the perspectives of approaching to the school algebra into the two parts. The one is the intrinsic nature of algebra, i.e. generalization, abstraction, and structural aspects. The other is the extrinsic representation of algebra, i.e. problem solving, modelling, functions. Each perspectives are investigated through various research studies. We hope that this works on each perspectives help a teacher prepare school algebra.

developed by solving practical problems and gradually formalized and abstracted. But school mathematics seemed to stress the formalized and abstracted mathematics. The same is the case with calculus. In particular, it appeared extremely in teaching of calculus. It caused hindrance of learning and indeed, many students had difficulties in teaming of calculus. Therefore this study investigates the various approaches of calculus teaching and the history of calculus which include approaches by Archimedes, Galileo, Newton, Leibniz and Weierstrass etc. This may offer the implication for calculus teaching and we can find the alternative on the method of calculus teaching in historico-genetic principle. Finally we suggest the direction of calculus teaching from this perspective in tile concrete.

This research's purpose is to investigate follows. 1. How do middle school teachers recognize the mathematical communication globally? 2. If we classify the modes of mathematical communication as written, spoken, graphic and active ones, how much do teachers use them and how do the students' communication ability come as teachers judge? 3. What are teachers' thinking, the present condition and the future indication for the application of mathematical communication with computer? 4. Do teachers evaluate their students' communication ability? If then, what is the assessment rubric of student's communication ability? The results are analyzed by frequency analysis including percentile and free writings are arranged by similar responses. The result of this study is that global recognition for mathematical communication, current state for students' concrete performance of mathematical communication, and assessment of mathematical communication & proposals are very lacking.

This paper is to make strides toward an enriched understanding of student learning and performance in mathematics that acknowledges the roles social and cultural contexts play in what students learn as well as what we are able to team about student learning. A student's mathematical practice over a year and a half is presented in detail in order to explore the relationships between classroom contexts and student performance. This study was situated at a K-4 urban elementary school in the United States. The data used for this study included classroom observations, interviews with the teachers and the student, and document collection. The data were analyzed by characterizing each classroom context and exploring the student's practice both in the classrooms and in the interviews. Despite the student's ongoing status as a struggling student, there were tremendous changes in his level of engagement in and persistence with mathematical tasks. The student was substantially more engaged in and enthusiastic about the daily mathematics lessons in third grade than he had been in second. However, we found little improvement in his mathematical understanding and performance during class or in the interviews. This highlights that increased engagement in the mathematical tasks does not necessarily signal increased learning. This paper discusses several issues of learning and performance raised by the student, looking at the relationship between classroom context and student performance. This paper also considers implications for how students' performances are interpreted and how learning is assessed.

In this paper, tangram-like puzzles which are made by dissecting square are introduced. Especially, tangram-like puzzles which are consists of five pieces, six pieces, seven pieces, eight pieces, nine pieces, ten pieces, twelve pieces, fourteen pieces are introduced. But, This Introduction is very superficial. It means introduction is focused on each piece's geometrical shape, relative area when each tangram-like puzzles' area is one. With this introduction, six tangram-like puzzles' implication in mathematics education are suggested as followings. (1) Tangram-like puzzles may help fostering spatial senses. (2) Tangram-like puzzles may help teaching polygons, and its properties, congruences, similarities, etc. (3)Tangram-like puzzles may help teaching additions of fractions. (4) Tangram-like puzzles may help fostering mathematical thinking. (5) Tangram-like puzzles may serve as topics for supplement or reinforcement in teaching and learning tangram. (6) Tangram-like puzzles may serve as topics for problem posing in teaching and learning tangram.

In this study, we studied some examples using GSP(Geometer's SketchPad) in the process of problem solving that is explained by G. polya. After reconsidering examples, we tried to show that using GSP can help student's intuitive thinking, investigative activities, reflective thinking. Especially, in the three phase of problem solving(understanding the problem, devising a plan, looking back), mathematics teachers may using GSP in order to helping student's understanding. Besides, we tried to suggest the direction to use GSP more adequately in the teaching and Beaming mathematics. First of all, Mathematics teachers using GSP in their class must have ideas how to use it. And they have to be careful on the didactical transposition of mathematical knowledge in the computer-based learning. They also have to lead students move from activities with GSP materials to carrying out the problem solving plan and reflection activities.